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Generalized theory of impact ionization in multilayered semiconductor structures
514
Surface Science 174 (lYX6) 514-518 North-Holland, Amsterdam
GENERALIZED THEORY OF IMPACT IONIZATION IN MULTILAYERED SEMICONDUCTOR STRUCTURES
Received
29 July lY85: accepted
for publication
12 Septemher
19X5
We present a general theory of impact ionization in superlattice avalanche photodiode (API)) devices particularly for those structures in which the carriers obtain no net energy gain from the superlattice itseif, the quantum well and undoped channeling APD. It is found that two mechantsms operate to enhance the electron ionization rate. The sudden large increase in kinetic energy derived from crossing a heterobarrier increases the probability of semiballistic impact ionization Simultaneously. the carriers are heated by the field while within the AlGaAs layer, such that upon x-entry into the GaAs layer the distribution is hotter than before. Carrier heating by the field 1s particularly important when the field is applied parallel to the layers as in the undoped channeling APD.
Recent experiments [1,2] show significant enhancement of the electron to hole ionization coefficients in the quantum well superlattice structure 131. We have performed Monte Carlo calculations which indicate that the electron ionization rate can indeed by strongly enhanced with respect to the hole ionization rate in the quantum well device. As shown in table 1, the electron ionization rate increases exponentially with increasing conduction band edge discontinuity, A EC [4], while the average distance an electron travels from the GaAs/AlGaAs interface before impact ionizing decreases monotonically. Both effects imply that electrons behave semiballistically as they cross Table 1 Quantum well device with constant applied _.-. Electron Wand edge ionization discontinuity rate (1 /cm) (eVf 0.08
0.14 0.24 0.347 0.42
X.0 X102 1.30x10” 227x10 4.14x 10’ 667x70~~
field of 250 kV/cm
(loo),
500 A well widths
Distance electron travels from interface before ionizing (A) 215.7
180.0 180.0 152.0 129.0
0039~6028/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) and Yamada Science Foundation
K. Brennan / Impact ionization in multilayered semiconductors
515
into the GaAs region from the AlGaAs layer. The boost in kinetic energy derived from AE, heats the electrons increasing the number within the high-energy tail of the distribution. This acts to enhance the impact ionization rate since the number of “lucky-drift” [5,6] electrons is increased. Theoretical calculations of carrier transport [7] in the channeling APD, an interdigitated p-n junction detector [8], indicate that optical device performance is obtained not when the electrons are highly confined to the GaAs layer and the holes swept out into the larger bandgap AlGaAs layers but rather when the carriers can drift freely from one layer to another. The electron ionization rate, CX,depends upon the presence of the potential step, AE,, and how frequently the interface is encountered, smaller layer widths [7]. However, the enhanced electron ionization rate in both the quantum well and undoped channeling APD seems unphysical on the basis of energy conservation. In either case, the net energy gain from the superlattice structure is zero. What then is the cause of the calculated increase in the electron ionization rate in these structures? Generally speaking the ionization rates, (Y and fi, in a structure with a spatially periodic electric field are enhanced above their respective values in the absence of the periodic modulation as a result of the strong non-linear (exponential) dependence of (Yand /3 on the electric field [9] and the existence of a threshold energy in the impact ionization process. Carriers moving in bulk semiconductor material under the influence of an applied electric field approach a quasi-steady-state energy distribution through the combined action of field heating and cooling from phonon emission. Eventually the carriers achieve steady state where the average energy gained from the field is balanced by energy lost to the lattice via phonon scattering. When a heterointerface is introduced, as in the undoped channeling APD, the carriers now may transfer into the AlGaAs layer from the GaAs. Upon transferring, the carriers are cooled by the band offset, AE, or AE,, below their quasi-steady-state average energy. The quasi-steady state is restored within the AlGaAs if the carriers gain more energy from the field on average than is lost to the phonons. Later, upon re-entering the GaAs, a carrier gains a kinetic energy boost equal to the band offset. Consequently, the carriers are now significantly hotter than they were before they transferred into the AlGaAs initially. In other words, the distribution is hotter within the GaAs because of the field heating in the AlGaAs layer. The above theory is substantiated by the calculations presented in fig. 1. The average energy gained from the field by electrons in the AlGaAs is plotted as a function of the conduction band edge discontinuity. We first calculate the average energy in the GaAs before the electrons transfer into the AlGaAs and the average energy in the AlGaAs before the electrons transfer into the GaAs. The net energy gained in the AlGaAs from the field is given by the difference between the conduction band offset and the difference in the above two average energies. Fig. 1 illustrates the results for an undoped channeling APD
K. Brenncm / impact ionrzation
516
in multila~yered semiconducror.v
.3,
I 0
0 GaAs Layer 0.1 pm A1,045Ga0,55A~ Layer 0.2 pm F = 250 kV/cm l
.
(100)
Quantum Well APD Barnet 84 Well .%ze = 5ooA F = 250 kV/cm (100)
. .
.i
.i
.i
Conduction
Fig. 1 Fieid heating quantum well APD
in the AlGaAs
.h
Band Edge Dlscontmwty
as a function
(eV)
of A&CCwithin
the channeling
APD and in the
structure at an applied electric field of 250 kV/cm and 0.1 pm GaAs and 0.2 pm AlGaAs layer widths. Clearly, field heating is very significant in the undoped channeling APD. The AlGaAs layers are modeied using a GaAs band structure with a modified energy gap and ionization threshold energy since an AlGaAs pseudopotential band structure is not available to us at the present time. Substitution of GaAs in place of AlGaAs may not be fully justified since in Al,,,,5Ga,,,As both r and L are degenerate. while in GaAs they are separated by approximately 0.30 eV. From this it can be argued that the electron distribution function should be cooler in the AlGaAs than in the GaAs since the phonon scattering rate will be dominated by deformation potential scattering due to the increased density of states at low energy. However, the density of states at and above the average carrier energies in these devices, - 0.70 eV, is probably the same in GaAs and AlGaAs. In addition. recent work by Blacha et al. [lo] suggests that the deformation potential constants are about the same in both GaAs and AlAs. Consequently, the phonon scattering rates should be comparable. The addition of alloy scattering may also act to cool the electron distribution in the AlGaAs, but we have found that the alloy scattering, as treated by Harrison et al. [ll], is an order of magnitude less than the deformation potential scattering at the energies of interest here. Interestingly, field heating in the AlGaAs is much less in the quantum well device than in the channeling APD as seen in fig. 1. However, as the AlGaAs layers increase in width, the field heating increases somewhat (fig. 2). This implies that the electrons must drift for some time in the AlGaAs layer in order to return to quasi-steady-state conditions. The question then is how does enhancement occur in the quantum well device? The reason is that the sudden
K. Brennan / Impact ionization in multilayered semiconductors
Band edge discontinuity
517
= .347eV
0 0
L
I 250
I 500
Barrier
Fig. 2. Field heating
in the AlGaAs
I 1000
I 750 Width
as a function
i
of barrier
width within the quantum
well APD.
change in kinetic energy provided by the heterointerface produces overshoot/undershoot behavior depending upon whether the carrier gains or loses energy. Consequently, when the electron enters the GaAs, it will overshoot its initial energy. When the electron re-enters the AlGaAs, it initially undershoots its steady-state energy. After the electrons drift for some time within the AlGaAs, the quasi-steady state is restored. In the undoped channeling APD, since the field is applied parallel to the layers, the electrons can drift for a long time before they return to the GaAs. The field heating can then be significant. In the quantum well APD, since the field is applied pe~endi~ular to the layer widths, the electrons drift within both the AlGaAs and GaAs layers for much shorter times before transferring. Therefore, the field heating in the AlGaAs is less significant. Finally, the question remains why is the hole ioni~tion rate not greatly enhanced? In the GaAs/AlGaAs material system the currently accepted values of AEc and AE, [12] indicate that the valence band discontinuity is significantly less than the conduction band discontinuity, A E, =:40% A Eg, such that the kinetic energy boost at the interface is much less for the holes. Therefore, the overshoot will be less significant. By this we mean that the energy distribution for holes is shifted on the average towards higher energy by the action of the potential discontinuity less than that of the electrons. Additionally, the hole energy relaxation rate is much larger than the electron relaxation rate for carrier energies less than 0.70 eV. Consequently, the holes are cooled more effectively than the electrons and, hence, the overshoot is reduced. Therefore, fewer holes drift to or above the ionization threshold energy resulting in fewer ionization events. This is consistent with recent experimental measurements which indicate that holes are more effectively captured in quantum wells than electrons [13,14]. In summa~, it is seen that the combined effect of field heating in the
51X
AlGaAs electron
K. Brennan
/ Impuct
roni:atwn
rn multilayered
semrconductorc
and undershoot/overshoot dynamics produces ionization rate in multilayered structures.
The author
would like to thank Dr.F. Capasso
enhancement
in the
for many helpful discussions.
References [l] F. Capasso. W.T. Tsang, A.L. Hutchinson and G.F. Williams, Appl. Phys. Letters 40 (1982) 3X. [2] F.-Y. Juang, Y. Nashimoto, P.K. Bhattacharya and S. Dhar, presented at the 43rd Device Research Conf., Boulder, CO. June 1985. [3] R. Chin. N. Holonyak. G.E. Stillman, J.Y. Tang and K. Hess. Electron. Letters 16 (1980) 467. [4] K. Brennan, T. Wang and K. Hess. IEEE Electron. Device Letters EDL-6 (1985) 199. [5] H. Shichijo and K. Hess, Phys. Rev. B23 (1981) 4197. [6] B.K. Ridley, J. Phys. Cl6 (7983) 3373. [7] K. Brennan, IEEE Trans. Electron Devices ED-32 (1985) 2197. [8] F. Capasso, IEEE Trans. Electron Devices ED-29 (1982) 1388. [9] K. Hess and F. Capasso, private communication. [lo] A. Blacha. H. Presting and M. Cardona, Phys. Status Solidi (b) 126 (1984) I I, [ll] J.W. Harrison and J.R. Hauser, Phys. Rev. Bl3 (1976) 5347. 1121 R.C. Miller, D.A. Kleinman and A.C. Gossard, Phys. Rev. B29 (1984) 70X5. [13] N. Holonyak. Jr.. R.M. Kotbas. R.D. Dupuis and P.D. Dapkus, IEEE J. Quantum Electron. QE-16 (1980) 170. [14] J.F. Ryan, R.A. Taylor, A.J. Turberfield. A. Maciel. J.M. Warlock, A.C. Gossard and W. Wiegmann. Phys. Rev. Letters 53 (1984) 1841.
Surface Science 174 (lYX6) 514-518 North-Holland, Amsterdam
GENERALIZED THEORY OF IMPACT IONIZATION IN MULTILAYERED SEMICONDUCTOR STRUCTURES
Received
29 July lY85: accepted
for publication
12 Septemher
19X5
We present a general theory of impact ionization in superlattice avalanche photodiode (API)) devices particularly for those structures in which the carriers obtain no net energy gain from the superlattice itseif, the quantum well and undoped channeling APD. It is found that two mechantsms operate to enhance the electron ionization rate. The sudden large increase in kinetic energy derived from crossing a heterobarrier increases the probability of semiballistic impact ionization Simultaneously. the carriers are heated by the field while within the AlGaAs layer, such that upon x-entry into the GaAs layer the distribution is hotter than before. Carrier heating by the field 1s particularly important when the field is applied parallel to the layers as in the undoped channeling APD.
Recent experiments [1,2] show significant enhancement of the electron to hole ionization coefficients in the quantum well superlattice structure 131. We have performed Monte Carlo calculations which indicate that the electron ionization rate can indeed by strongly enhanced with respect to the hole ionization rate in the quantum well device. As shown in table 1, the electron ionization rate increases exponentially with increasing conduction band edge discontinuity, A EC [4], while the average distance an electron travels from the GaAs/AlGaAs interface before impact ionizing decreases monotonically. Both effects imply that electrons behave semiballistically as they cross Table 1 Quantum well device with constant applied _.-. Electron Wand edge ionization discontinuity rate (1 /cm) (eVf 0.08
0.14 0.24 0.347 0.42
X.0 X102 1.30x10” 227x10 4.14x 10’ 667x70~~
field of 250 kV/cm
(loo),
500 A well widths
Distance electron travels from interface before ionizing (A) 215.7
180.0 180.0 152.0 129.0
0039~6028/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) and Yamada Science Foundation
K. Brennan / Impact ionization in multilayered semiconductors
515
into the GaAs region from the AlGaAs layer. The boost in kinetic energy derived from AE, heats the electrons increasing the number within the high-energy tail of the distribution. This acts to enhance the impact ionization rate since the number of “lucky-drift” [5,6] electrons is increased. Theoretical calculations of carrier transport [7] in the channeling APD, an interdigitated p-n junction detector [8], indicate that optical device performance is obtained not when the electrons are highly confined to the GaAs layer and the holes swept out into the larger bandgap AlGaAs layers but rather when the carriers can drift freely from one layer to another. The electron ionization rate, CX,depends upon the presence of the potential step, AE,, and how frequently the interface is encountered, smaller layer widths [7]. However, the enhanced electron ionization rate in both the quantum well and undoped channeling APD seems unphysical on the basis of energy conservation. In either case, the net energy gain from the superlattice structure is zero. What then is the cause of the calculated increase in the electron ionization rate in these structures? Generally speaking the ionization rates, (Y and fi, in a structure with a spatially periodic electric field are enhanced above their respective values in the absence of the periodic modulation as a result of the strong non-linear (exponential) dependence of (Yand /3 on the electric field [9] and the existence of a threshold energy in the impact ionization process. Carriers moving in bulk semiconductor material under the influence of an applied electric field approach a quasi-steady-state energy distribution through the combined action of field heating and cooling from phonon emission. Eventually the carriers achieve steady state where the average energy gained from the field is balanced by energy lost to the lattice via phonon scattering. When a heterointerface is introduced, as in the undoped channeling APD, the carriers now may transfer into the AlGaAs layer from the GaAs. Upon transferring, the carriers are cooled by the band offset, AE, or AE,, below their quasi-steady-state average energy. The quasi-steady state is restored within the AlGaAs if the carriers gain more energy from the field on average than is lost to the phonons. Later, upon re-entering the GaAs, a carrier gains a kinetic energy boost equal to the band offset. Consequently, the carriers are now significantly hotter than they were before they transferred into the AlGaAs initially. In other words, the distribution is hotter within the GaAs because of the field heating in the AlGaAs layer. The above theory is substantiated by the calculations presented in fig. 1. The average energy gained from the field by electrons in the AlGaAs is plotted as a function of the conduction band edge discontinuity. We first calculate the average energy in the GaAs before the electrons transfer into the AlGaAs and the average energy in the AlGaAs before the electrons transfer into the GaAs. The net energy gained in the AlGaAs from the field is given by the difference between the conduction band offset and the difference in the above two average energies. Fig. 1 illustrates the results for an undoped channeling APD
K. Brenncm / impact ionrzation
516
in multila~yered semiconducror.v
.3,
I 0
0 GaAs Layer 0.1 pm A1,045Ga0,55A~ Layer 0.2 pm F = 250 kV/cm l
.
(100)
Quantum Well APD Barnet 84 Well .%ze = 5ooA F = 250 kV/cm (100)
. .
.i
.i
.i
Conduction
Fig. 1 Fieid heating quantum well APD
in the AlGaAs
.h
Band Edge Dlscontmwty
as a function
(eV)
of A&CCwithin
the channeling
APD and in the
structure at an applied electric field of 250 kV/cm and 0.1 pm GaAs and 0.2 pm AlGaAs layer widths. Clearly, field heating is very significant in the undoped channeling APD. The AlGaAs layers are modeied using a GaAs band structure with a modified energy gap and ionization threshold energy since an AlGaAs pseudopotential band structure is not available to us at the present time. Substitution of GaAs in place of AlGaAs may not be fully justified since in Al,,,,5Ga,,,As both r and L are degenerate. while in GaAs they are separated by approximately 0.30 eV. From this it can be argued that the electron distribution function should be cooler in the AlGaAs than in the GaAs since the phonon scattering rate will be dominated by deformation potential scattering due to the increased density of states at low energy. However, the density of states at and above the average carrier energies in these devices, - 0.70 eV, is probably the same in GaAs and AlGaAs. In addition. recent work by Blacha et al. [lo] suggests that the deformation potential constants are about the same in both GaAs and AlAs. Consequently, the phonon scattering rates should be comparable. The addition of alloy scattering may also act to cool the electron distribution in the AlGaAs, but we have found that the alloy scattering, as treated by Harrison et al. [ll], is an order of magnitude less than the deformation potential scattering at the energies of interest here. Interestingly, field heating in the AlGaAs is much less in the quantum well device than in the channeling APD as seen in fig. 1. However, as the AlGaAs layers increase in width, the field heating increases somewhat (fig. 2). This implies that the electrons must drift for some time in the AlGaAs layer in order to return to quasi-steady-state conditions. The question then is how does enhancement occur in the quantum well device? The reason is that the sudden
K. Brennan / Impact ionization in multilayered semiconductors
Band edge discontinuity
517
= .347eV
0 0
L
I 250
I 500
Barrier
Fig. 2. Field heating
in the AlGaAs
I 1000
I 750 Width
as a function
i
of barrier
width within the quantum
well APD.
change in kinetic energy provided by the heterointerface produces overshoot/undershoot behavior depending upon whether the carrier gains or loses energy. Consequently, when the electron enters the GaAs, it will overshoot its initial energy. When the electron re-enters the AlGaAs, it initially undershoots its steady-state energy. After the electrons drift for some time within the AlGaAs, the quasi-steady state is restored. In the undoped channeling APD, since the field is applied parallel to the layers, the electrons can drift for a long time before they return to the GaAs. The field heating can then be significant. In the quantum well APD, since the field is applied pe~endi~ular to the layer widths, the electrons drift within both the AlGaAs and GaAs layers for much shorter times before transferring. Therefore, the field heating in the AlGaAs is less significant. Finally, the question remains why is the hole ioni~tion rate not greatly enhanced? In the GaAs/AlGaAs material system the currently accepted values of AEc and AE, [12] indicate that the valence band discontinuity is significantly less than the conduction band discontinuity, A E, =:40% A Eg, such that the kinetic energy boost at the interface is much less for the holes. Therefore, the overshoot will be less significant. By this we mean that the energy distribution for holes is shifted on the average towards higher energy by the action of the potential discontinuity less than that of the electrons. Additionally, the hole energy relaxation rate is much larger than the electron relaxation rate for carrier energies less than 0.70 eV. Consequently, the holes are cooled more effectively than the electrons and, hence, the overshoot is reduced. Therefore, fewer holes drift to or above the ionization threshold energy resulting in fewer ionization events. This is consistent with recent experimental measurements which indicate that holes are more effectively captured in quantum wells than electrons [13,14]. In summa~, it is seen that the combined effect of field heating in the
51X
AlGaAs electron
K. Brennan
/ Impuct
roni:atwn
rn multilayered
semrconductorc
and undershoot/overshoot dynamics produces ionization rate in multilayered structures.
The author
would like to thank Dr.F. Capasso
enhancement
in the
for many helpful discussions.
References [l] F. Capasso. W.T. Tsang, A.L. Hutchinson and G.F. Williams, Appl. Phys. Letters 40 (1982) 3X. [2] F.-Y. Juang, Y. Nashimoto, P.K. Bhattacharya and S. Dhar, presented at the 43rd Device Research Conf., Boulder, CO. June 1985. [3] R. Chin. N. Holonyak. G.E. Stillman, J.Y. Tang and K. Hess. Electron. Letters 16 (1980) 467. [4] K. Brennan, T. Wang and K. Hess. IEEE Electron. Device Letters EDL-6 (1985) 199. [5] H. Shichijo and K. Hess, Phys. Rev. B23 (1981) 4197. [6] B.K. Ridley, J. Phys. Cl6 (7983) 3373. [7] K. Brennan, IEEE Trans. Electron Devices ED-32 (1985) 2197. [8] F. Capasso, IEEE Trans. Electron Devices ED-29 (1982) 1388. [9] K. Hess and F. Capasso, private communication. [lo] A. Blacha. H. Presting and M. Cardona, Phys. Status Solidi (b) 126 (1984) I I, [ll] J.W. Harrison and J.R. Hauser, Phys. Rev. Bl3 (1976) 5347. 1121 R.C. Miller, D.A. Kleinman and A.C. Gossard, Phys. Rev. B29 (1984) 70X5. [13] N. Holonyak. Jr.. R.M. Kotbas. R.D. Dupuis and P.D. Dapkus, IEEE J. Quantum Electron. QE-16 (1980) 170. [14] J.F. Ryan, R.A. Taylor, A.J. Turberfield. A. Maciel. J.M. Warlock, A.C. Gossard and W. Wiegmann. Phys. Rev. Letters 53 (1984) 1841.